unity blender with armature set in blender
2 things to try:

A) Unity and most game engines are sensitive to having a root position to base all other transformations off of.  Unless you tell it otherwise, that root transform will be the bone at the top of the armature hierarchy.  Make sure you have a root bone that all other bones are parented to, and make sure you do not apply animations to that bone.  Without knowing more, I suspect you only have one bone, are animating it, and Unity is not handling that very well.

(EDIT) With the file provided, it turns out A is mostly true, but there is one more aspect you need to address.  If you check the console of Unity, when it imports, it actually gives a descriptive error: 
This means that all the verticies, which have not weight associated for them to a bone, will default to the first bone (the lid).  So moving the lid will move the rest of the box too...because they were arbitrarily assigned to the lid.  Just weight paint the rest of the box onto your root bone, and it works just fine.  

B) It sometimes helps conceptually to have a one or two frame "idle" animation for things, especially if you are using the Mecanim editor in Unity.  Then you have better control as to when the animation plays and how.